In Romeo and Juliet, act 2, scene 3, why does the friar agree to marry the two young lovers?

Quick answer:

Friar Laurence is at first confused by Romeo's request to marry Juliet, since, the day before, Romeo had been complaining to him about the fact that Rosaline would not reciprocate his undying love. However, Friar Laurence agrees to marry the two young lovers, mostly in the hopes that this secret marriage will end the lengthy Capulet-Montague feud. (His plan does ultimately end the feud, but certainly not in the peaceful, painless way he was expecting.)

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As other answers have noted, the friar states that he agrees to the hasty and secret marriage in the hopes it will end the feud between the two families. He understands, the level of feuding has lately been heating up, to the point that the prince has declared death for any Capulet or Montague caught fighting on the streets of Verona.
The friar, as it happens, has been pondering issues of good and evil as he tends to the herbs in his garden. Many of his plants are both medicinal and poisonous, depending on how they are applied. As he says to himself just before Romeo arrives:
Virtue itself turns vice, being misapplied,
And vice sometime by action dignified.
The friar hopes that the "vice" or seeming wrong of secretly marrying the two young lovers without the approval of their parents will work for the good.
The friar also clearly has a long and intimate history with Romeo, who, we find, comes to visit him often. He knows all about Romeo's lovesick woes over Rosaline (this is quite a contrast to Romeo's father, who is so mystified by Romeo's mooning that he asks Benvolio what is going on). The friar is, in fact, so shocked by Romeo's turnaround that he initially can hardly believe what Romeo is saying to him, berating him for having cried so many tears over Rosaline only to abandon her as soon as Juliet came along.
Reading between the lines and knowing that the friar knows Romeo so well, we might infer that he realizes it is futile to oppose the match. Romeo is the son of a rich and powerful lord and wants what he wants when he wants it. It is very likely the friar is trying to rationalize what he knows he will be bullied into anyway, just as Romeo later bullies the apothecary into selling him poison. But what we can say for fact is that the friar tells Romeo:
In one respect I'll thy assistant be;
For this alliance may so happy prove,
To turn your households' rancour to pure love.
Knowing Romeo well, the friar also advises him to go slowly. This Romeo, of course, will not do, although the friar says he should act:

Wisely and slow. They stumble that run fast.

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In Act 2, Scene 3, Romeo asks Friar Laurence to secretly marry Juliet and him.  At first, the friar does not want to perform the wedding—rightly so, too, because the day before Romeo had told Friar Laurence how sick he was over the fact that his dear Rosaline would rather be celibate than be with Romeo.  

Thou and these woes were all for Rosaline:
And art thou changed?

Romeo convinces the friar that he is no longer in love with Rosaline, because he is very, truly in love with Juliet.  

Then plainly know my heart's dear love is set
On the fair daughter of rich Capulet:
As mine on hers, so hers is set on mine;
And all combined, save what thou must combine
By holy marriage

Friar Laurence does then agree to perform the secret wedding ceremony for Romeo and Juliet.  What is interesting about the friar's agreement is his reason for performing the marriage ceremony.  He never once says that he will do it because Romeo and Juliet are so in love with each other.  Friar Laurence agrees to marry the two star crossed lovers because he thinks that the marriage will perhaps end the feud between the Montagues and the Capulets.  

But come, young waverer, come, go with me,
In one respect I'll thy assistant be;
For this alliance may so happy prove,
To turn your households' rancour to pure love.

I have a hard time faulting the friar.  His intentions are well placed.  His plan makes sense.  But as Robert Burns wrote in "To a Mouse,"

“the best laid plans of mice and men often go awry.”

And for sure did Friar Laurence's plan go awry.  

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Romeo goes and asks Friar Laurence to marry him and Juliet. At first Friar Laurence is confused to why Romeo wants to marry Juliet. He thinks Romeo is still in love with Rosaline. He questions Romeo about this, but Romeo convinces the friar that he and Juliet are truly in love, so the friar agrees to marry them. 

Friar Laurence agrees to marry the two young lovers, because he believes that it will cause an end to the grudge between the two families. The Montagues and the Capulets have hated each other for some time. When Romeo and Juliet fall in love with each other, Friar Laurence believes this is the answer for there to be peace within the two families. Friar Laurence wants both of the families to be at peace with each other, and he thinks this could be the perfect way for this to be accomplished, so he goes ahead and marries them.

When Friar Laurence learns of the death of Tybalt at the hands of Romeo, he is the one who comes up with the plan for Juliet to drink the potion to make it look like she has died, so he can have time to get to Romeo and tell him of the plan. Of course, this plan backfires, because his message does not get to Romeo in time. In an ironic twist, the death of the two could bring the families together like the marriage should have. Friar Laurence has a big hand in both of the most important decisions of the entire play. 

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"I'll thy assistant be;
For this alliance may so happy prove
To turn your households’ rancour to pure love."(95)

Romeo asks Friar Lawrence to marry Juliet and him even though their families are enemies. The Friar is confused at first and against the idea.  He asks Romeo what happened to his professed love for Rosaline.  Romeo finally convinces the Friar to agree to wed the two young lovers in the hopes that their love for each other, and their marriage, would end the hatred the two families feel toward each other.

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